Statistik 2

Hochschule Aalen

Sommersemester 2021

Martin Heckmann

martin.heckmann@hs-aalen.de

19.05.2021

Assignment 4: Parameter Estimation

Target

Target of the assignment is to understand the principles of parameter estimation.

Task 1

• As previously, randomly extract a sample of $N$ data points from the data set americans.csv
• Vary the size of the sample $N$ from $ N \in \{10, 50, 100, 500, 1000\}$ data points
• Repeat this for each sample size $N$ 1000 times and calculate mean, median, mode and standard deviation of the height and weight of the americans from these data points, i.e. calculate 1000 times mean, median, mode and standard deviation for samples of size $N=10$, then 1000 times for samples of size $N=50$, ...
• Store the results (5*1000 means, medians, modes, standard deviations) in a pandas data frame
• Also calculate mean, median, mode and standard deviation from the full data set (population mean, median, ...) and store it
• Visualize the results as separate histograms (one for height sample size 10, one for 50, ...) using the Plotly function graph_objects.Histogram. Overlay the histograms for the means, medians and modes in one histogram using opacity=0.75 as follows:

  fig = go.Figure()
  fig.add_trace(go.Histogram(x= ...
  fig.add_trace(go.Histogram(x= ...
  ...
  fig.update_traces(opacity=0.75, ...
  fig.update_layout(
    title='Height'
  )
  fig.show()
  `
  

• Add the population statistics (mean, median, ...) and the statistics from the 1000 runs of the experiment to the plots (mean, median, ... calculated from the 1000 values for for $N=10$, $N=50$, ...)
• Make sure you initialize the seed of the random generator at the beginning of your code such that you have identical results each time you run the code
• Select each sample of $N$ points "truly" at random and not by selecting consecutive chunks of $N$ data points. The 1000 sets of $N$ data points do not need to be disjoint, i.e. the same data point can be part of many of the 1000 sets.

Task 2

• For $N=100$ calculate the distribution of men in the data sets, i.e. count in each data set you drew how many men it contains, divide it be $N$ and show a histogram of these values.
• Add the ratio of men in the full data set, i.e. the population
• Repeat task 1 for men and women separately
• Compare the results (histograms with statistics) for $N=100$ when you used men and women, only men and only women

Task 3

• For $N=50$ and $N=500$ perform a linear regression of the weight using the height as a feature
• In each case, draw 1000 samples of size $N=50$, $N=500$ respectively and randomly subdivide them into a training set with $70\%$ of the data and a test with the remaining $30\%$.
• Visualize the distribution of the paramters $w_0$ and $w_1$ in two separate ($N=50$, $N=500$) 2D histograms. Calculate the mean and the variance of the parameters. Add the mean to the histograms. Also add the parameter values you obtain when you train on $70\%$ of the full data set and train on the remaining $30\%$ (population values).
• For each case, calculate the mean squared error of the regression. In a single histogram, show the histograms of the errors, the means of the errors and the error on the full data.
• You can use the following code snippets as starting points ```python # Splitting the data in train and test X_train, X_test, y_train, y_test = train_test_split( ... X, y, test_size=0.3, random_state=0)

Package for linear regression

from sklearn.linear_model import LinearRegression

Package to calculate performance metrics

from sklearn import metrics

We init the model

lin_reg = LinearRegression()

We perform the fit by providing the X matrix and the y vector

Note: We do not need to prepend 1 to X as this is done internally

lin_reg.fit(X_train, y_train)

We print the coefficients (w0 = intercept, w_1 = linrg.coef)

print(linreg.intercept, linreg.coef)

Predict the test data

y_predict = lin_reg.predict(X_test)

Calculate the residuals, i.e. errors after fit

MSE = metrics.mean_squared_error(y_test, y_predict)

Drawing several histograms in one plot

bin_width = 1 fig = plotly.graph_objects.Figure() fig.add_trace(plotly.graph_objects.Histogram(x=data_frame[data_frame['size']==size1, name=size1,xbins=dict(size=bin_width))) fig.add_trace(plotly.graph_objects.Histogram(x=data_frame[data_frame['size']==size2, name=size2, xbins=dict(size=bin_width)))

fig.add_scatter(x=[population_value,population_value],y=[0, hist_height],name="population value")

fig.update_traces(opacity=0.75) fig.update_layout( title="MSE", barmode='overlay' ) fig.show()

#Task1
#As previously, randomly extract a sample of 𝑁 data points from the data set americans.csv
import pandas
import numpy
import plotly.graph_objects 
import plotly.express 

numpy.random.seed(31)

americans=pandas.read_csv("americans.csv",delimiter=",")
americans

	Unnamed: 0	CaseID	Age	Sex	BMI	Height (in)	Weight (lb)	Height (cm)	Weight (kg)	DoesGroceries
0	0	20160101160045	62	2	26.6	66	165	167.64	74.910	True
1	1	20160101160066	69	1	44.3	69	300	175.26	136.200	True
2	2	20160101160069	24	2	24.5	64	143	162.56	64.922	True
3	3	20160101160083	31	2	21.2	57	98	144.78	44.492	True
4	4	20160101160084	59	2	29.7	64	173	162.56	78.542	True
...	...	...	...	...	...	...	...	...	...	...
9693	10205	20161212162307	52	2	33.6	65	202	165.10	91.708	False
9694	10206	20161212162357	69	1	26.5	68	174	172.72	78.996	True
9695	10207	20161212162426	28	1	27.1	64	158	162.56	71.732	True
9696	10208	20161212162456	80	2	44.9	65	270	165.10	122.580	False
9697	10209	20161212162509	16	2	22.8	68	150	172.72	68.100	False

9698 rows × 10 columns

dataframes=[]
N=[10,50,100,500,1000]
for i in range(len(N)):        
    #calculate mean, median, mode and standard deviation of the height and weight 
    mean_h,mean_w,median_h,median_w,mode_w,mode_h,std_h,std_w,men_ratio=[],[],[],[],[],[],[],[],[]
    for j in range(1000):
        sample=americans.sample(n=N[i])
        mean_h.append(sample['Height (cm)'].mean())
        mean_w.append(sample['Weight (kg)'].mean())
        median_h.append(sample['Height (cm)'].median())
        median_w.append(sample['Weight (kg)'].median())        
        mode_h.append(sample['Height (cm)'].mode()[0])
        mode_w.append(sample['Weight (kg)'].mode()[0])
        std_h.append(sample['Height (cm)'].std())
        std_w.append(sample['Weight (kg)'].std())
        try:
            men_ratio.append(sample['Sex'].value_counts().get(1)/N[i])
        except:
            men_ratio.append(0.0)
    df=pandas.DataFrame()
    df['mean_h']=mean_h
    df['mean_w']=mean_w
    df['median_h']=median_h
    df['median_w']=mean_w
    df['mode_h']=mode_h
    df['mode_w']=mode_w
    df['std_h']=std_h
    df['std_w']=std_w
    df['men_ratio']=men_ratio
    dataframes.append(df)
    
#dataframes[4]

F_mean_h=americans['Height (cm)'].mean()
F_median_h=americans['Height (cm)'].median()
F_mode_h=americans['Height (cm)'].mode()[0]
F_std_h=americans['Height (cm)'].std()
F_mean_w=americans['Weight (kg)'].mean()
F_median_w=americans['Weight (kg)'].median()
F_mode_w=americans['Weight (kg)'].mode()[0]
F_std_w=americans['Weight (kg)'].std()

for i in range(len(N)): 
    #max_=dataframes[i]['mean_h'].max()
    max_=max(dataframes[i]['mean_h'].max(),dataframes[i]['median_h'].max(),dataframes[i]['mode_h'].max())
    min_=min(dataframes[i]['mean_h'].min(),dataframes[i]['median_h'].min(),dataframes[i]['mode_h'].min())
    #min_=dataframes[i]['mean_h'].min()
    #print(sampleResults[i].count())
    bin_width=(max_-min_)/21
    #print(bin_width)
    fig = plotly.graph_objects.Figure()
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='blue'),x=dataframes[i]['mean_h'], name='mean',xbins=dict(size=bin_width)))
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='orange'),x=dataframes[i]['median_h'], name='median',xbins=dict(size=bin_width)))
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='green'),x=dataframes[i]['mode_h'], name='mode',xbins=dict(size=bin_width)))
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='purple'),x=dataframes[i]['std_h'], name='std',xbins=dict(size=(dataframes[i]['std_h'].max()-dataframes[i]['std_h'].min())/21),visible = 'legendonly'))
    
    N_mean=dataframes[i]['mean_h'].mean()
    N_median=dataframes[i]['median_h'].median()
    N_mode=dataframes[i]['mode_h'].mode()[0]
    N_std=dataframes[i]['std_h'].std()
    fig.add_vline(line=dict(color='blue'),x=N_mean,name="sample_mean", annotation=dict(text='sample_mean', showarrow=True))
    fig.add_vline(line=dict(color='orange'),x=N_median,name="sample_mean", annotation=dict(yshift=-80,text='sample_median', showarrow=True))
    fig.add_vline(line=dict(color='green'),x=N_mode,name="sample_mean", annotation=dict(yshift=-160,text='sample_mode', showarrow=True))
    #fig.add_vline(line=dict(color='purple'),x=N_std,name="sample_mean", annotation=dict(text='sample_std', showarrow=True))
    fig.add_vline(line=dict(color='blue'),x=F_mean_h,name="sample_mean", annotation=dict(yshift=-20,text='full_mean', showarrow=True))
    fig.add_vline(line=dict(color='orange'),x=F_median_h,name="sample_mean", annotation=dict(yshift=-100,text='full_median', showarrow=True))
    fig.add_vline(line=dict(color='green'),x=F_mode_h,name="sample_mean", annotation=dict(yshift=-180,text='full_mode', showarrow=True))
    #fig.add_vline(line=dict(color='purple'),x=F_std_h,name="sample_mean", annotation=dict(text='full_std', showarrow=True))
    #fig.add_scatter(marker=dict(color='blue'),x=[N_mean,N_mean],y=[0,270],name="sample_mean")
    #fig.add_scatter(marker=dict(color='orange'),x=[N_median,N_median],y=[0,270],name="sample_median")
    #fig.add_scatter(marker=dict(color='green'),x=[N_mode,N_mode],y=[0,270],name="sample_mode")
    fig.add_scatter(marker=dict(color='purple'),x=[N_std,N_std],y=[0,270],name="sample_std",visible='legendonly')
    #fig.add_scatter(marker=dict(color='blue'),x=[F_mean_h,F_mean_h],y=[0,270],name="full_mean")
    #fig.add_scatter(marker=dict(color='orange'),x=[F_median_h,F_median_h],y=[0,270],name="full_median")
    #fig.add_scatter(marker=dict(color='green'),x=[F_mode_h,F_mode_h],y=[0,270],name="full_mode")
    fig.add_scatter(marker=dict(color='purple'),x=[F_std_h,F_std_h],y=[0,270],name="full_std",visible='legendonly')

    fig.update_traces(opacity=0.5)
    fig.update_layout(
        title=str(N[i])+" Heigth (cm)",
        barmode='overlay'
    )
    fig.show()
    max_=max(dataframes[i]['mean_w'].max(),dataframes[i]['median_w'].max(),dataframes[i]['mode_w'].max())
    min_=min(dataframes[i]['mean_w'].min(),dataframes[i]['median_w'].min(),dataframes[i]['mode_w'].min())
    #min_=dataframes[i]['mean_h'].min()
    #print(sampleResults[i].count())
    #bin_width=(max_-min_)/21
    bin_width=1
    #print(bin_width)
    fig = plotly.graph_objects.Figure()
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='blue'),x=dataframes[i]['mean_w'], name='mean',xbins=dict(size=bin_width)))
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='orange'),x=dataframes[i]['median_w'], name='median',xbins=dict(size=bin_width)))
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='green'),x=dataframes[i]['mode_w'], name='mode',xbins=dict(size=bin_width)))
    fig.add_trace(plotly.graph_objects.Histogram(marker=dict(color='purple'),x=dataframes[i]['std_w'], name='std',xbins=dict(size=(dataframes[i]['std_w'].max()-dataframes[i]['std_w'].min())/21),visible = 'legendonly'))
    
    N_mean=dataframes[i]['mean_w'].mean()
    N_median=dataframes[i]['median_w'].median()
    N_mode=dataframes[i]['mode_w'].mode()[0]
    N_std=dataframes[i]['std_w'].std()
    
    fig.add_vline(line=dict(color='blue'),x=N_mean,name="sample_mean", annotation=dict(text='sample_mean', showarrow=True))
    fig.add_vline(line=dict(color='orange'),x=N_median,name="sample_mean", annotation=dict(yshift=-80,text='sample_median', showarrow=True))
    fig.add_vline(line=dict(color='green'),x=N_mode,name="sample_mean", annotation=dict(yshift=-160,text='sample_mode', showarrow=True))
    fig.add_vline(line=dict(color='blue'),x=F_mean_w,name="sample_mean", annotation=dict(yshift=-20,text='full_mean', showarrow=True))
    fig.add_vline(line=dict(color='orange'),x=F_median_w,name="sample_mean", annotation=dict(yshift=-100,text='full_median', showarrow=True))
    fig.add_vline(line=dict(color='green'),x=F_mode_w,name="sample_mean", annotation=dict(yshift=-180,text='full_mode', showarrow=True))
    fig.add_scatter(marker=dict(color='purple'),x=[N_std,N_std],y=[0,270],name="sample_std",visible='legendonly')            
    fig.add_scatter(marker=dict(color='purple'),x=[F_std_w,F_std_w],y=[0,270],name="full_std",visible='legendonly')
    
    #fig.add_scatter(marker=dict(color='blue'),x=[F_mean_h,F_mean_h],y=[0,270],name="full_mean")
    #fig.add_scatter(marker=dict(color='orange'),x=[F_median_h,F_median_h],y=[0,270],name="full_median")
    #fig.add_scatter(marker=dict(color='green'),x=[F_mode_h,F_mode_h],y=[0,270],name="full_mode")
    #fig.add_scatter(marker=dict(color='purple'),x=[F_std_h,F_std_h],y=[0,270],name="full_std",visible='legendonly')
    #fig.add_scatter(marker=dict(color='blue'),x=[N_mean,N_mean],y=[0,270],name="sample_mean")
    #fig.add_scatter(marker=dict(color='blue'),x=[F_mean_w,F_mean_w],y=[0,270],name="full_mean")
    #fig.add_scatter(marker=dict(color='orange'),x=[N_median,N_median],y=[0,270],name="sample_median")
    #fig.add_scatter(marker=dict(color='orange'),x=[F_median_w,F_median_w],y=[0,270],name="full_median")
    #fig.add_scatter(marker=dict(color='green'),x=[N_mode,N_mode],y=[0,270],name="sample_mode")
    #fig.add_scatter(marker=dict(color='green'),x=[F_mode_w,F_mode_w],y=[0,270],name="full_mode")
    #
    #fig.add_scatter(marker=dict(color='purple'),x=[F_std_w,F_std_w],y=[0,270],name="full_std",visible='legendonly')

    fig.update_traces(opacity=0.5)    
    fig.update_layout(
        title=str(N[i])+" Weight (kg)",
        barmode='overlay'
    )
    fig.show()
#Task1 fin

df_n100=dataframes[2]
fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['men_ratio']))
fig.add_vline(x=americans['Sex'].value_counts().get(1)/len(americans))
fig.show()

americans_men = americans[americans['Sex'] == 1]
americans_women = americans[americans['Sex'] == 2]
mean_h,mean_w,median_h,median_w,mode_w,mode_h,std_h,std_w=[],[],[],[],[],[],[],[]
for j in range(1000):
    sample=americans_men.sample(n=100)
    mean_h.append(sample['Height (cm)'].mean())
    mean_w.append(sample['Weight (kg)'].mean())
    median_h.append(sample['Height (cm)'].median())
    median_w.append(sample['Weight (kg)'].median())        
    mode_h.append(sample['Height (cm)'].mode()[0])
    mode_w.append(sample['Weight (kg)'].mode()[0])
    std_h.append(sample['Height (cm)'].std())
    std_w.append(sample['Weight (kg)'].std())
df_m=pandas.DataFrame()
df_m['mean_h']=mean_h
df_m['mean_w']=mean_w
df_m['median_h']=median_h
df_m['median_w']=mean_w
df_m['mode_h']=mode_h
df_m['mode_w']=mode_w
df_m['std_h']=std_h
df_m['std_w']=std_w


mean_h,mean_w,median_h,median_w,mode_w,mode_h,std_h,std_w=[],[],[],[],[],[],[],[]
for j in range(1000):
    sample=americans_women.sample(n=100)
    mean_h.append(sample['Height (cm)'].mean())
    mean_w.append(sample['Weight (kg)'].mean())
    median_h.append(sample['Height (cm)'].median())
    median_w.append(sample['Weight (kg)'].median())        
    mode_h.append(sample['Height (cm)'].mode()[0])
    mode_w.append(sample['Weight (kg)'].mode()[0])
    std_h.append(sample['Height (cm)'].std())
    std_w.append(sample['Weight (kg)'].std())
df_w=pandas.DataFrame()
df_w['mean_h']=mean_h
df_w['mean_w']=mean_w
df_w['median_h']=median_h
df_w['median_w']=mean_w
df_w['mode_h']=mode_h
df_w['mode_w']=mode_w
df_w['std_h']=std_h
df_w['std_w']=std_w


fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['mean_h'], name='men & women',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['mean_h'], name='men',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['mean_h'], name='women',xbins=dict(size=1)))
fig.update_traces(opacity=.75)
fig.update_layout(
    title="Height mean compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['mean_w'], name='men & women',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['mean_w'], name='men',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['mean_w'], name='women',xbins=dict(size=1)))
fig.update_traces(opacity=.75)
fig.update_layout(
    title="Weight mean compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['median_h'], name='men & women',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['median_h'], name='men',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['median_h'], name='women',xbins=dict(size=1)))
fig.update_traces(opacity=.75)
fig.update_layout(
    title="Height median compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['median_w'], name='men & women',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['median_w'], name='men',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['median_w'], name='women',xbins=dict(size=1)))
fig.update_traces(opacity=.75)
fig.update_layout(
    title="Weight median compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['mode_h'], name='men & women',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['mode_h'], name='men',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['mode_h'], name='women',xbins=dict(size=1)))
fig.update_traces(opacity=.5)
fig.update_layout(
    title="Height mode compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['mode_w'], name='men & women',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['mode_w'], name='men',xbins=dict(size=1)))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['mode_w'], name='women',xbins=dict(size=1)))
fig.update_traces(opacity=.5)
fig.update_layout(
    title="Weight mode compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['std_h'], name='men & women'))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['std_h'], name='men'))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['std_h'], name='women'))
fig.update_traces(opacity=.75)
fig.update_layout(
    title="Height std compared",
    barmode='overlay'
)
fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=df_n100['std_w'], name='men & women'))
fig.add_trace(plotly.graph_objects.Histogram(x=df_m['std_w'], name='men'))
fig.add_trace(plotly.graph_objects.Histogram(x=df_w['std_w'], name='women'))
fig.update_traces(opacity=.75)
fig.update_layout(
    title="Weight std compared",
    barmode='overlay'
)
fig.show()
#Task2 fin

from sklearn.linear_model import LinearRegression
from sklearn import metrics
from sklearn.model_selection import train_test_split
erg=pandas.DataFrame()
N=[50,500]
for i in range(len(N)):
    w0,w1,mse=[],[],[]
    for j in range(1000):
        sample=americans.sample(n=N[i])
        x_train, x_test, y_train, y_test = train_test_split(sample['Height (cm)'], sample['Weight (kg)'],test_size=.3, random_state=0)
        lin_reg = LinearRegression()
        #Reshape your data either using array.reshape(-1, 1) if your data has a single feature or array.reshape(1, -1) if it contains a single sample.
        x_train=x_train.values.reshape(-1,1)
        #fit(X training data, y target values, sample_weight=None) https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression.fit
        lin_reg.fit(x_train, y_train)
        #print(lin_reg.intercept_, lin_reg.coef_[0])
        w0.append(lin_reg.intercept_)
        w1.append(lin_reg.coef_[0])
        
        #MSE
        y_predict = lin_reg.predict(x_test.values.reshape(-1,1))
        mse.append(metrics.mean_squared_error(y_test, y_predict))
        
    erg[''+str(N[i])+'_w0']=w0
    erg[''+str(N[i])+'_w1']=w1
    erg[''+str(N[i])+'_mse']=mse
    #print(''+str(N[i])+'_mse')

#whole populatation 70% training 30%test
x_train, x_test, y_train, y_test = train_test_split(sample['Height (cm)'], sample['Weight (kg)'],test_size=.3, random_state=0)
pop_lin_reg = LinearRegression()
x_train=x_train.values.reshape(-1,1)
pop_lin_reg.fit(x_train, y_train)
y_predict = pop_lin_reg.predict(x_test.values.reshape(-1,1))
pop_mse=metrics.mean_squared_error(y_test, y_predict)

for i in range(len(N)):
    fig = plotly.graph_objects.Figure()
    fig.add_trace(plotly.graph_objects.Histogram(x=erg[''+str(N[i])+'_w0'], name='w0'))
    fig.add_trace(plotly.graph_objects.Histogram(x=erg[''+str(N[i])+'_w1'], name='w1'))
    z=erg[''+str(N[i])+'_w0'].mean()
    fig.add_trace(plotly.graph_objects.Scatter(x=[z,z],y=[0,80], name='w0 mean', mode='lines',marker=dict(color='#636EFA' )))
    z=erg[''+str(N[i])+'_w1'].mean()
    fig.add_trace(plotly.graph_objects.Scatter(x=[z,z],y=[0,80], name='w1 mean', mode='lines',marker=dict(color='#EF553B' )))
    z=erg[''+str(N[i])+'_w0'].std()
    fig.add_trace(plotly.graph_objects.Scatter(x=[z,z],y=[0,80], name='w0 std', mode='lines',visible='legendonly'))
    z=erg[''+str(N[i])+'_w1'].std()
    fig.add_trace(plotly.graph_objects.Scatter(x=[z,z],y=[0,80], name='w1 std', mode='lines',visible='legendonly'))
    z=pop_lin_reg.intercept_
    fig.add_trace(plotly.graph_objects.Scatter(x=[z,z],y=[0,80], name='wo pop', mode='lines',marker=dict(color='#4F5FF7' )))
    z=pop_lin_reg.coef_[0]
    fig.add_trace(plotly.graph_objects.Scatter(x=[z,z],y=[0,80], name='w1 pop', mode='lines',marker=dict(color='#ED3012' )))
    fig.update_traces(opacity=.75)
    fig.update_layout(
        title='n='+str(N[i]),
        barmode='overlay'
    )
    fig.show()

fig = plotly.graph_objects.Figure()
fig.add_trace(plotly.graph_objects.Histogram(x=erg['50_mse'], name='mse n=50'))
fig.add_trace(plotly.graph_objects.Histogram(x=erg['500_mse'], name='mse n=500'))
z=erg['50_mse'].mean()
fig.add_trace(plotly.graph_objects.Scatter(x=[z,z], y=[0, 110], name='mse mean n=50', mode='lines',marker=dict(color='#4F5FF7' )))
z=erg['500_mse'].mean()
fig.add_trace(plotly.graph_objects.Scatter(x=[z, z], y=[0, 110], name='mse mean n=500', mode='lines',marker=dict(color='#ED3012' )))
fig.add_trace(plotly.graph_objects.Scatter(x=[pop_mse, pop_mse], y=[0, 120], name='mse pop', mode='lines'))

fig.update_traces(opacity=.75)
fig.update_layout(
    title='MSE',
    barmode='overlay'
)
fig.show()